Markov property

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for a real-valued stochastic process ,

The property that for any set of times from and any Borel set ,


with probability 1, that is, the conditional probability distribution of given coincides (almost certainly) with the conditional distribution of given . This can be interpreted as independence of the "future" and the "past" given the fixed "present" . Stochastic processes satisfying the property (*) are called Markov processes (cf. Markov process). The Markov property has (under certain additional assumptions) a stronger version, known as the "strong Markov property" . In discrete time the strong Markov property, which is always true for (Markov) sequences satisfying (*), means that for each stopping time (relative to the family of -algebras , ), with probability one


[1] I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian)



[a1] K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960)
[a2] J.L. Doob, "Stochastic processes" , Wiley (1953)
[a3] E.B. Dynkin, "Markov processes" , 1 , Springer (1965) (Translated from Russian)
[a4] T.G. Kurtz, "Markov processes" , Wiley (1986)
[a5] W. Feller, "An introduction to probability theory and its applications" , 1–2 , Wiley (1966)
[a6] P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965)
[a7] M. Loève, "Probability theory" , II , Springer (1978)
How to Cite This Entry:
Markov property. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article